Why Eve gets caught

A walkthrough of all four basis-choice cases
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1 · Alice's choice

Pick a basis and bit for Alice. The same Alice photon is sent into all four cases below.

Basis
Bit

2 · Pick a case to step through

Eve guesses one of two bases (she doesn't know Alice's). Bob also picks a basis. There are four combinations:

3 · Walk through the optics

ALICE EVE BOB 405 nm laser Alice λ/2 Alice→ + Eve λ/2 PBS det 0 det 1 resend Eve λ/2 Eve→ + Bob λ/2 PBS det 0 det 1 P = ½ P = ½ P = ½ P = ½
Pick Alice's basis + bit, then pick a case above.
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4 · Run this case 1000 times — what fraction does Bob get right?

When Eve's basis matches Alice's, her measurement is deterministic and Bob's outcome (when his basis also matches) is correct. When Eve's basis disagrees with Alice's, her measurement is random — and that's what corrupts the bit.

Pick a case first.
Trials
Bob = Alice (correct)
Bob ≠ Alice (error)
Error rate